Open Access

Existence results for nonlinear boundary value problems with integral boundary conditions on an infinite interval

Boundary Value Problems20122012:127

DOI: 10.1186/1687-2770-2012-127

Received: 29 June 2012

Accepted: 22 October 2012

Published: 5 November 2012

Abstract

In this paper, by using fixed point theorems in a cone, the existence of one positive solution and three positive solutions for nonlinear boundary value problems with integral boundary conditions on an infinite interval are established.

MSC:34B10, 39A10, 34B18, 45G10.

Keywords

positive solutions fixed point theorems integral boundary conditions infinite interval

1 Introduction

Consider the following boundary value problem with integral boundary conditions on the half-line of an infinite interval of the form
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ1_HTML.gif
(1.1)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ2_HTML.gif
(1.2)

where f C ( ( 0 , ) × [ 0 , ) × R , [ 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq1_HTML.gif, f may be singular at t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq2_HTML.gif; g 1 , g 2 : [ 0 , ) [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq3_HTML.gif are continuous, nondecreasing functions and for 0 t < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq4_HTML.gif, z in a bounded set, g 1 ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq5_HTML.gif, g 2 ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq6_HTML.gif are bounded; ψ : [ 0 , ) ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq7_HTML.gif is a continuous function with 0 ψ ( s ) d s < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq8_HTML.gif; p C [ 0 , ) C 1 ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq9_HTML.gif with p ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq10_HTML.gif on ( 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq11_HTML.gif and 0 + d s p ( s ) < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq12_HTML.gif; a 1 + a 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq13_HTML.gif, b i > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq14_HTML.gif for i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq15_HTML.gif with D = a 2 b 1 + a 1 b 2 + a 1 a 2 B ( 0 , ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq16_HTML.gif in which B ( t , s ) = t s d s p ( s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq17_HTML.gif.

Boundary value problems on an infinite interval appear often in applied mathematics and physics. There are many papers concerning the existence of solutions on the half-line for boundary value problems; see [15] and the references therein.

At the same time, boundary value problems with integral boundary conditions are of great importance and are an interesting class of problems. They constitute two, three, multi-point, and nonlocal boundary value problems as special cases. For an overview of the literature on integral boundary value problems, see [611] and the references therein.

Yan Sun et al. [4] studied the existence of positive solutions for singular boundary value problems on the half-line for the following Sturm-Liouville boundary value problem:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equa_HTML.gif
where μ is a positive parameter; f is a continuous, non-negative function and may be singular at t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq2_HTML.gif; p C [ 0 , ) C 1 ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq18_HTML.gif with p ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq10_HTML.gif on ( 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq11_HTML.gif and 0 + d s p ( s ) < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq12_HTML.gif; a i , b i 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq19_HTML.gif for i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq20_HTML.gif. Wang et al. [5] investigated the existence theorems for the boundary value problem given by
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equb_HTML.gif
where f is a continuous, non-negative function and may be singular at t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq2_HTML.gif; p C [ 0 , ) C 1 ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq21_HTML.gif with p ( t ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq10_HTML.gif on ( 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq11_HTML.gif and 0 + d s p ( s ) < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq12_HTML.gif; α i β i 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq22_HTML.gif for i = 1 , 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq20_HTML.gif. Also, Feng [11] considered the following boundary value problem with integral boundary conditions on a finite interval:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equc_HTML.gif

where a , b > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq23_HTML.gif; g C 1 ( [ 0 , 1 ] , [ 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq24_HTML.gif, w L p [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq25_HTML.gif, 1 p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq26_HTML.gif, and h L 1 [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq27_HTML.gif are symmetric functions; f : [ 0 , 1 ] × [ 0 , ) [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq28_HTML.gif is continuous. The author obtained the existence of symmetric positive solutions by using the fixed point index theory in cones.

Motivated by the above works, we consider the existence of one and three positive solutions for the BVP (1.1), (1.2). However, to our knowledge, although various existence theorems are obtained for Sturm-Liouville boundary value problems with homogeneous boundary conditions, problems with nonhomogeneous boundary conditions, especially integral boundary conditions on an infinite interval have rarely been considered. Therefore, our boundary conditions are more general.

The rest of the paper is organized as follows. In Section 2, we present some necessary lemmas that will be used to prove our main results. In Section 3, we apply the Schauder fixed point theorem to get the existence of at least one positive solution for the nonlinear boundary value problem (1.1) and (1.2). In Section 4, we use the Leggett-Williams fixed point theorem [12] to get the existence of at least three positive solutions for the nonlinear boundary value problem (1.1) and (1.2).

2 Preliminaries

In this section, we will employ several lemmas to prove the main results in this paper. These lemmas are based on the following BVP for h C ( ( 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq29_HTML.gif:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ3_HTML.gif
(2.1)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ4_HTML.gif
(2.2)
Define φ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq30_HTML.gif and θ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq31_HTML.gif to be the solutions of the corresponding homogeneous equation
1 p ( t ) ( p ( t ) z ( t ) ) = 0 , t ( 0 , ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ5_HTML.gif
(2.3)
under the initial conditions,
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ6_HTML.gif
(2.4)
Using the initial conditions (2.4), we can deduce, from equation (2.3) for θ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq31_HTML.gif and φ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq30_HTML.gif, the following equations:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ7_HTML.gif
(2.5)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ8_HTML.gif
(2.6)
Let G ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq32_HTML.gif be the Green function for (2.1), (2.2) is given by
G ( t , s ) = 1 D { θ ( t ) φ ( s ) , 0 t s < ; θ ( s ) φ ( t ) , 0 s t < , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ9_HTML.gif
(2.7)

where θ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq31_HTML.gif and φ ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq30_HTML.gif are given in (2.5) and (2.6) respectively.

Lemma 2.1 Suppose the conditions 0 + d s p ( s ) < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq33_HTML.gif and D > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq34_HTML.gif hold. Then for any h C ( ( 0 , ) , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq35_HTML.gif, the BVP (2.1), (2.2) has the unique solution
z ( t ) = 0 G ( t , s ) p ( s ) h ( s ) d s + φ ( t ) D 0 g 1 ( z ( s ) ) ψ ( s ) d s + θ ( t ) D 0 g 2 ( z ( s ) ) ψ ( s ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equd_HTML.gif

where G ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq32_HTML.gif is given by (2.7).

Furthermore, it is easy to prove the following properties of G ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq32_HTML.gif:
  1. (1)

    G ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq32_HTML.gif is continuous on [ 0 , + ) × [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq36_HTML.gif.

     
  2. (2)

    For each s [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq37_HTML.gif, G ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq32_HTML.gif is continuously differentiable on [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq38_HTML.gif except t = s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq39_HTML.gif.

     
  3. (3)

    G ( t , s ) t | t = s + G ( t , s ) t | t = s = 1 p ( s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq40_HTML.gif.

     
  4. (4)
    | G ( t , s ) | c p ( t ) G ( s , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq41_HTML.gif, for t [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq42_HTML.gif, where
    c = max { a 1 , a 2 } min { b 1 , b 2 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ10_HTML.gif
    (2.8)
     
  5. (5)

    For each s [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq37_HTML.gif, G ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq32_HTML.gif satisfies the corresponding homogeneous BVP (i.e., h ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq43_HTML.gif in the BVP (2.1)) on [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq38_HTML.gif except t = s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq39_HTML.gif.

     
  6. (6)
    0 G ( t , s ) G ( s , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq44_HTML.gif for t , s [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq45_HTML.gif and
    G ¯ ( s ) : = lim t + G ( t , s ) = b 2 D θ ( s ) G ( s , s ) < + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Eque_HTML.gif
     
  7. (7)
    For any t [ a , b ] ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq46_HTML.gif and s [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq37_HTML.gif, we have
    G ( t , s ) γ 0 G ( s , s ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equf_HTML.gif
     
where
γ 0 = min { b 2 + a 2 B ( b , ) b 2 + a 2 B ( 0 , ) , b 1 + a 1 B ( 0 , a ) b 1 + a 1 B ( 0 , ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equg_HTML.gif

Obviously, 0 < γ 0 < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq47_HTML.gif.

It is convenient to list the following conditions which are to be used in our theorems:

(H1) f C ( ( 0 , ) × [ 0 , ) × R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq48_HTML.gif and also, u ( t ) h ( x , y ) f ( t , x , y ) v ( t ) h ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq49_HTML.gif, 0 < t < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq50_HTML.gif, where h C ( [ 0 , ) × R , ( 0 , ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq51_HTML.gif; and for 0 t < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq4_HTML.gif, x, y in a bounded set, h ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq52_HTML.gif is bounded and u , v : ( 0 , ) ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq53_HTML.gif is continuous and may be singular at t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq2_HTML.gif; and also, there exists 0 < k 0 < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq54_HTML.gif such that u ( t ) k 0 v ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq55_HTML.gif for t ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq56_HTML.gif.

(H2) g 1 , g 2 : [ 0 , ) [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq3_HTML.gif are continuous, nondecreasing functions, and for 0 t < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq4_HTML.gif, z in a bounded set, g 1 ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq5_HTML.gif, g 2 ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq6_HTML.gif are bounded.

(H3) ψ : [ 0 , ) ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq57_HTML.gif is a continuous function with 0 ψ ( s ) d s < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq58_HTML.gif.

(H4) 0 G ( s , s ) p ( s ) v ( s ) d s < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq59_HTML.gif and 0 G ( s , s ) p ( s ) u ( s ) d s < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq60_HTML.gif.

Consider the Banach space
B = { z C 1 [ 0 , + ) : lim t + z ( t ) < , sup t [ 0 , ) | z ( t ) | < } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equh_HTML.gif

with the norm z = max { sup t [ 0 , ) | z ( t ) | , sup t [ 0 , ) | z ( t ) | } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq61_HTML.gif.

From the above assumptions, we can define an operator A : B B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq62_HTML.gif by
A z ( t ) = 0 G ( t , s ) p ( s ) f ( s , z ( s ) , z ( s ) ) d s + φ ( t ) D 0 g 1 ( z ( s ) ) ψ ( s ) d s + θ ( t ) D 0 g 2 ( z ( s ) ) ψ ( s ) d s , t ( 0 , ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ11_HTML.gif
(2.9)

where G ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq32_HTML.gif is given by (2.7).

Lemma 2.2 ([13])

Let be defined as before and M B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq63_HTML.gif. Then M is relatively compact in if the following conditions hold:
  1. (a)

    M is uniformly bounded in ;

     
  2. (b)

    The functions belonging to M are equicontinuous on any compact interval of [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq64_HTML.gif;

     
  3. (c)

    The functions from M are equiconvergent, that is, given ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq65_HTML.gif, there corresponds a T ( ϵ ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq66_HTML.gif such that | f ( t ) f ( + ) | < ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq67_HTML.gif for any t T ( ϵ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq68_HTML.gif and f M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq69_HTML.gif.

     

Definition 2.1 An operator is called completely continuous if it is continuous and maps bounded sets into relatively compact sets.

3 Existence of at least one positive solution

In this section, we will apply the following Schauder fixed point theorem to get an existence of one positive solution.

Theorem 3.1 (Schauder fixed point theorem)

Let be a Banach space and S be a nonempty bounded, convex, and closed subset of . Assume A : B B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq70_HTML.gif is a completely continuous operator. If the operator A leaves the set S invariant, i.e., if A ( S ) S https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq71_HTML.gif, then A has at least one fixed point in S.

For convenience, let us set
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equi_HTML.gif
and
B ( R ) = M D [ g 1 ( R ) + g 2 ( R ) ] 0 ψ ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equj_HTML.gif
Theorem 3.2 Assume conditions (H1)-(H4) are satisfied. In addition, let there exist a number R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq72_HTML.gif such that
max { 1 , sup t [ 0 , ) c p ( t ) } sup z 1 [ 0 , R ] , z 2 [ R , R ] h ( z 1 , z 2 ) ( 0 G ( s , s ) p ( s ) v ( s ) d s ) + B ( R ) R , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equk_HTML.gif

where c is defined by (2.8).

Then the BVP (1.1), (1.2) has at least one solution z with
0 z ( t ) R , t [ 0 , ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equl_HTML.gif
Proof Let A : B B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq62_HTML.gif be the operator defined by (2.9). We claim that A is a completely continuous operator. To justify this, we first show that A : B B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq62_HTML.gif is well defined. Let z B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq73_HTML.gif, then there exists r 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq74_HTML.gif such that z r 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq75_HTML.gif and from conditions (H1) and (H2), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equm_HTML.gif
and
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equn_HTML.gif
Let t 1 , t 2 [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq76_HTML.gif, t 1 < t 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq77_HTML.gif, then
0 | G ( t 1 , s ) G ( t 2 , s ) | p ( s ) v ( s ) d s 2 0 G ( s , s ) p ( s ) v ( s ) d s < + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ12_HTML.gif
(3.1)
Hence, by the Lebesgue dominated convergence theorem and the fact that G ( t , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq32_HTML.gif is continuous on t, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ13_HTML.gif
(3.2)
Also, by (H1) and (H2), we get
| ( A z ) ( t 1 ) ( A z ) ( t 2 ) | a 2 D | 1 p ( t 1 ) 1 p ( t 2 ) | 0 t 1 θ ( s ) p ( s ) f ( s , z ( s ) , z ( s ) ) d s + a 1 D p ( t 1 ) t 1 t 2 φ ( s ) p ( s ) f ( s , z ( s ) , z ( s ) ) d s + a 1 D | 1 p ( t 1 ) 1 p ( t 2 ) | t 2 φ ( s ) p ( s ) f ( s , z ( s ) , z ( s ) ) d s + a 2 D p ( t 2 ) t 1 t 2 θ ( s ) p ( s ) f ( s , z ( s ) , z ( s ) ) d s + a 2 D | 1 p ( t 1 ) 1 p ( t 2 ) | 0 g 1 ( z ( s ) ) ψ ( s ) d s + a 1 D | 1 p ( t 1 ) 1 p ( t 2 ) | 0 g 2 ( z ( s ) ) ψ ( s ) d s a 2 S r 0 D | 1 p ( t 1 ) 1 p ( t 2 ) | 0 t 1 θ ( s ) p ( s ) v ( s ) d s + a 1 S r 0 D p ( t 1 ) t 1 t 2 φ ( s ) p ( s ) v ( s ) d s + a 1 S r 0 D | 1 p ( t 1 ) 1 p ( t 2 ) | t 2 φ ( s ) p ( s ) v ( s ) d s + a 2 S r 0 D p ( t 2 ) t 1 t 2 θ ( s ) p ( s ) v ( s ) d s + a 2 T r 0 D | 1 p ( t 1 ) 1 p ( t 2 ) | 0 ψ ( s ) d s + a 1 T r 0 D | 1 p ( t 1 ) 1 p ( t 2 ) | 0 ψ ( s ) d s 0 as  t 1 t 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ14_HTML.gif
(3.3)

So, A z C 1 [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq78_HTML.gif.

We can show that A z B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq79_HTML.gif. Notice that
lim t + ( A z ) ( t ) = 0 G ¯ ( s ) p ( s ) f ( s , z ( s ) , z ( s ) ) d s + φ ( ) D 0 g 1 ( z ( s ) ) ψ ( s ) d s + θ ( ) D 0 g 2 ( z ( s ) ) ψ ( s ) d s < + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equo_HTML.gif
In addition, we have
| ( A z ) ( t ) | 0 | G ( t , s ) | p ( s ) v ( s ) h ( z ( s ) , z ( s ) ) d s + a 2 D p ( t ) 0 g 1 ( z ( s ) ) ψ ( s ) d s + a 1 D p ( t ) 0 g 2 ( z ( s ) ) ψ ( s ) d s c p ( t ) 0 G ( s , s ) p ( s ) v ( s ) h ( z ( s ) , z ( s ) ) d s + max { a 1 , a 2 } D p ( t ) 0 [ g 1 ( z ( s ) ) + g 2 ( z ( s ) ) ] ψ ( s ) d s S r 0 c p ( t ) 0 G ( s , s ) p ( s ) v ( s ) d s + ( T r 0 + T r 0 ) max { a 1 , a 2 } D p ( t ) 0 ψ ( s ) d s . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equp_HTML.gif

Therefore, sup | ( A z ) ( t ) | < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq80_HTML.gif.

Hence, A : B B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq62_HTML.gif is well defined.

Next, for any positive integer m, we denote the operator A m : B B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq81_HTML.gif by
( A m z ) ( t ) = 1 m G ( t , s ) p ( s ) f ( s , z ( s ) , z ( s ) ) d s + φ ( t ) D 0 g 1 ( z ( s ) ) ψ ( s ) d s + θ ( t ) D 0 g 2 ( z ( s ) ) ψ ( s ) d s , t [ 0 , ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ15_HTML.gif
(3.4)
and prove that A m : B B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq81_HTML.gif is completely continuous for each m 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq82_HTML.gif. Let z n z 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq83_HTML.gif as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq84_HTML.gif. We will show that A m z n A m z 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq85_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq86_HTML.gif in . We know that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equq_HTML.gif

where r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq87_HTML.gif is a real number such that r max n N { z , z n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq88_HTML.gif, N is a natural number set, S r : = sup { h ( x , y ) : 0 x r , r y r } < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq89_HTML.gif.

Therefore, for any ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq90_HTML.gif, there exists a sufficiently large K 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq91_HTML.gif ( K 0 > 1 m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq92_HTML.gif) such that
2 S r K 0 G ( s , s ) p ( s ) v ( s ) d s < ε 4 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ16_HTML.gif
(3.5)
From the fact that z n z 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq83_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq86_HTML.gif, we can see that for the above ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq90_HTML.gif, there exists a sufficiently large natural number N 0 N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq93_HTML.gif such that if n > N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq94_HTML.gif, for any s [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq95_HTML.gif, we have
| z n ( s ) z ( s ) | z n z < ε 4 ( M D 1 m K 0 G ( k , k ) p ( k ) d k ) 1 ( 0 ψ ( s ) d s ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equr_HTML.gif
and
| z n ( s ) z ( s ) | z n z < ε 4 ( M D 1 m K 0 G ( k , k ) p ( k ) d k ) 1 ( 0 ψ ( s ) d s ) 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equs_HTML.gif
On the other hand, by the continuity of f ( t , x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq96_HTML.gif, for the above ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq90_HTML.gif, there exists a δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq97_HTML.gif, for any t [ 1 m , K 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq98_HTML.gif, x , x 1 [ 0 , r ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq99_HTML.gif, y , y 1 [ r , r ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq100_HTML.gif such that if | x x 1 | < δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq101_HTML.gif, | y y 1 | < δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq102_HTML.gif, we have
| f ( t , x , y ) f ( t , x 1 , y 1 ) | < ε 4 ( 1 m K 0 G ( k , k ) p ( k ) d k ) 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ17_HTML.gif
(3.6)
From the fact that z n z 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq83_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq86_HTML.gif, there exists a natural number N 1 > N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq103_HTML.gif such that when n > N 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq104_HTML.gif, for any s [ 1 m , K 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq105_HTML.gif, z n ( s ) , z ( s ) [ 0 , r ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq106_HTML.gif, z n ( s ) , z ( s ) [ r , r ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq107_HTML.gif if | z n ( s ) z ( s ) | < δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq108_HTML.gif, | z n ( s ) z ( s ) | < δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq109_HTML.gif, we have
| f ( s , z n ( s ) , z n ( s ) ) f ( s , z ( s ) , z ( s ) ) | < ε 4 ( 1 m K 0 G ( k , k ) p ( k ) d k ) 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ18_HTML.gif
(3.7)
In addition to this, by the continuity of g 1 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq110_HTML.gif and g 2 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq111_HTML.gif on [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq112_HTML.gif, for the above ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq90_HTML.gif, there exists a δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq97_HTML.gif for any t [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq113_HTML.gif, x , x 1 [ 0 , r ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq99_HTML.gif, such that if | x x 1 | < δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq101_HTML.gif, we have
| g i ( x ) g i ( x 1 ) | < ε 4 ( M D 0 ψ ( s ) d s ) 1 , for  i = 1 , 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ19_HTML.gif
(3.8)
From z n z 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq83_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq86_HTML.gif, there exists a natural number N 2 > N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq114_HTML.gif such that when n > N 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq115_HTML.gif, for any s [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq95_HTML.gif, z n ( s ) , z ( s ) [ 0 , r ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq106_HTML.gif if | z n ( s ) z ( s ) | < δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq108_HTML.gif, we have
| g i ( z n ( s ) ) g i ( z ( s ) ) | < ε 4 ( M D 0 ψ ( s ) d s ) 1 , for  i = 1 , 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ20_HTML.gif
(3.9)
Hence, if N = max { N 1 , N 2 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq116_HTML.gif, then
| ( A m z n ) ( t ) ( A m z ) ( t ) | = | 1 m G ( t , s ) p ( s ) f ( s , z n ( s ) , z n ( s ) ) d s 1 m G ( t , s ) p ( s ) f ( s , z ( s ) , z ( s ) ) d s + φ ( t ) D 0 [ g 1 ( z n ( s ) ) g 1 ( z ( s ) ) ] ψ ( s ) d s + θ ( t ) D 0 [ g 2 ( z n ( s ) ) g 2 ( z ( s ) ) ] ψ ( s ) d s | 1 m K 0 G ( s , s ) p ( s ) | f ( s , z n ( s ) , z n ( s ) ) f ( s , z ( s ) , z ( s ) ) | d s + K 0 G ( s , s ) p ( s ) ( f ( s , z n ( s ) , z n ( s ) ) + f ( s , z ( s ) , z ( s ) ) ) d s + φ ( 0 ) D 0 | g 1 ( z n ( s ) ) g 1 ( z ( s ) ) | ψ ( s ) d s + θ ( ) D 0 | g 2 ( z n ( s ) ) g 2 ( z ( s ) ) | ψ ( s ) d s 1 m K 0 G ( s , s ) p ( s ) | f ( s , z n ( s ) , z n ( s ) ) f ( s , z ( s ) , z ( s ) ) | d s + 2 S r K 0 G ( s , s ) p ( s ) v ( s ) d s + M D 0 | g 1 ( z n ( s ) ) g 1 ( z ( s ) ) | ψ ( s ) d s + M D 0 | g 2 ( z n ( s ) ) g 2 ( z ( s ) ) | ψ ( s ) d s = ε . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equt_HTML.gif

Similarly, we can see that when z n z 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq83_HTML.gif as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq84_HTML.gif, | ( A m z n ) ( t ) ( A m z ) ( t ) | 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq117_HTML.gif as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq84_HTML.gif. This implies that A m : B B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq118_HTML.gif is a continuous operator for each natural number m.

Choose P R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq119_HTML.gif to be a bounded, convex, and closed set by
PR = { z B : z R , z ( t ) 0  for each  t [ 0 , ) } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equu_HTML.gif

We must show that there exists a positive constant R such that for each z P R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq120_HTML.gif, one has A z R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq121_HTML.gif.

Let z PR https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq122_HTML.gif. Then for each t [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq113_HTML.gif, we have G ( t , s ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq123_HTML.gif. Since f, g 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq124_HTML.gif, g 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq125_HTML.gif are positive functions, A m z ( t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq126_HTML.gif, t [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq113_HTML.gif. Furthermore, for t [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq113_HTML.gif
| ( A m z ) ( t ) | sup z 1 [ 0 , R ] , z 2 [ R , R ] h ( z 1 , z 2 ) ( 1 m G ( s , s ) p ( s ) v ( s ) d s ) + φ ( 0 ) D 0 g 1 ( z ( s ) ) ψ ( s ) d s + θ ( ) D 0 g 2 ( z ( s ) ) ψ ( s ) d s sup z 1 [ 0 , R ] , z 2 [ R , R ] h ( z 1 , z 2 ) ( 1 m G ( s , s ) p ( s ) v ( s ) d s ) + M D [ g 1 ( R ) + g 2 ( R ) ] 0 ψ ( s ) d s sup z 1 [ 0 , R ] , z 2 [ R , R ] h ( z 1 , z 2 ) ( 0 G ( s , s ) p ( s ) v ( s ) d s ) + B ( R ) R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ21_HTML.gif
(3.10)
and
| ( A m z ) ( t ) | 1 m | G ( t , s ) | p ( s ) v ( s ) h ( z ( s ) , z ( s ) ) d s + a 2 D p ( t ) 0 g 1 ( z ( s ) ) ψ ( s ) d s + a 1 D p ( t ) 0 g 2 ( z ( s ) ) ψ ( s ) d s c p ( t ) sup z 1 [ 0 , R ] , z 2 [ R , R ] h ( z 1 , z 2 ) 0 G ( s , s ) p ( s ) v ( s ) d s + max { a 1 , a 2 } D sup t [ 0 , ) 1 p ( t ) [ g 1 ( R ) + g 2 ( R ) ] 0 ψ ( s ) d s sup t [ 0 , ) c p ( t ) [ sup z 1 [ 0 , R ] , z 2 [ R , R ] h ( z 1 , z 2 ) 0 G ( s , s ) p ( s ) v ( s ) d s + M D [ g 1 ( R ) + g 2 ( R ) ] 0 ψ ( s ) d s ] R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ22_HTML.gif
(3.11)
Inequalities (3.10) and (3.11) yield that A m z R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq127_HTML.gif. Hence, A m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq128_HTML.gif is uniformly bounded. Using the similar proof as (3.2) and (3.3), we can obtain that for any t , t 1 [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq129_HTML.gif, z P R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq120_HTML.gif,
A m z ( t ) A m z ( t 1 ) 0 as  t t 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equv_HTML.gif
Thus, A m P R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq130_HTML.gif is equicontinuous. It follows from
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ23_HTML.gif
(3.12)
and
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ24_HTML.gif
(3.13)

Therefore, A m P R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq130_HTML.gif is equiconvergent. Hence, by Lemma 2.2 and the above discussion, we conclude that for each natural number m, A m : P P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq131_HTML.gif is completely continuous.

Finally, observe that
| ( A z ) ( t ) ( A m z ) ( t ) | = | 0 1 m G ( t , s ) p ( s ) f ( s , z ( s ) , z ( s ) ) d s | S r 0 0 1 m G ( s , s ) p ( s ) v ( s ) d s < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ25_HTML.gif
(3.14)
and
| ( A z ) ( t ) ( A m z ) ( t ) | = | 0 1 m G ( t , s ) p ( s ) f ( s , z ( s ) , z ( s ) ) d s | S r 0 sup t [ 0 , ) c p ( t ) 0 1 m G ( s , s ) p ( s ) v ( s ) d s < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ26_HTML.gif
(3.15)
Hence, inequalities (3.14) and (3.15) imply that sup t [ 0 , ) | ( A z ) ( t ) ( A m z ) ( t ) | < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq132_HTML.gif and sup t [ 0 , ) | ( A z ) ( t ) ( A m z ) ( t ) | < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq133_HTML.gif. Then by the assumption (H4) and the absolute continuity of the integral, we get
lim m 0 1 m G ( t , s ) p ( s ) v ( s ) d s = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equw_HTML.gif

Therefore, the operator A : B B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq62_HTML.gif is completely continuous and maps the set P R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq119_HTML.gif into itself. Hence, the Schauder fixed point theorem can be applied to obtain a solution of the BVP (1.1), (1.2). The theorem is proved. □

Example 3.1 Consider the following boundary value problem:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ27_HTML.gif
(3.16)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ28_HTML.gif
(3.17)

where f ( t , z ( t ) , z ( t ) ) = e 2 t ( 1 + t ) 12 t ( z + | z | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq134_HTML.gif, a 1 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq135_HTML.gif, a 2 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq136_HTML.gif, b 1 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq137_HTML.gif, b 2 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq138_HTML.gif, p ( t ) = e t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq139_HTML.gif, ψ ( s ) = 1 1 + s 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq140_HTML.gif, g 1 ( z ( s ) ) = g 2 ( z ( s ) ) = z ( s ) 24 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq141_HTML.gif.

It is clear that f : ( 0 , ) × [ 0 , + ) × R ( 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq142_HTML.gif is continuous and singular at t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq2_HTML.gif. Set v ( t ) = e 2 t ( 1 + t ) t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq143_HTML.gif and h ( z , z ) = z + | z | 12 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq144_HTML.gif, it follows from a direct calculation that M = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq145_HTML.gif, c = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq146_HTML.gif, and there exists R = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq147_HTML.gif such that the following inequality holds:
max { 1 , sup t [ 0 , ) 1 e t } sup z 1 [ 0 , 1 ] , z 2 [ 1 , 1 ] h ( z 1 , z 2 ) ( 0 G ( s , s ) p ( s ) v ( s ) d s ) + B ( 1 ) 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equx_HTML.gif

Then by Theorem 3.2, the boundary value problem (3.16)-(3.17) has at least one positive solution.

4 Existence of at least three positive solutions

Definition 4.1 Let be a Banach space, P B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq148_HTML.gif be a cone in . By a concave nonnegative continuous functional φ on P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq149_HTML.gif, we mean a continuous functional φ : P [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq150_HTML.gif with
φ ( t x + ( 1 t ) y ) t φ ( x ) + ( 1 t ) φ ( y ) for all  x , y P  and  t [ 0 , 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equy_HTML.gif
For K , L , R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq151_HTML.gif being constants with P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq149_HTML.gif and φ as above, let
P K = { x P : x < K } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equz_HTML.gif
and
P ( φ , L , K ) = { x P : L φ ( x ) , x K } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equaa_HTML.gif

Theorem 4.1 (Leggett-Williams fixed point theorem [12])

Let be a Banach space, P B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq148_HTML.gif be a cone of , and R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq72_HTML.gif be a constant. Suppose A : P ¯ R P ¯ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq152_HTML.gif is a completely continuous operator and φ is a nonnegative, continuous, concave functional on P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq149_HTML.gif with φ ( y ) y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq153_HTML.gif for all y P ¯ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq154_HTML.gif. If there exist r, L, and K with 0 < r < L < K R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq155_HTML.gif such that the following conditions hold:
  1. (i)

    { y P ( φ , L , K ) : φ ( y ) > L } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq156_HTML.gif and φ ( A y ) > L https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq157_HTML.gif for all y P ( φ , L , K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq158_HTML.gif;

     
  2. (ii)

    A y < r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq159_HTML.gif for all y r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq160_HTML.gif;

     
  3. (iii)

    φ ( A y ) > L https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq157_HTML.gif for all y P ( φ , L , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq161_HTML.gif with A y > K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq162_HTML.gif.

     
Then A has at least three positive solutions y 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq163_HTML.gif, y 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq164_HTML.gif, and y 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq165_HTML.gif in P ¯ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq166_HTML.gif satisfying
y 1 < r , y 2 { y P ( φ , L , R ) : φ ( y ) > L } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equab_HTML.gif
and
y 3 P ¯ R { P ( φ , L , R ) P ¯ r } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equac_HTML.gif

Theorem 4.2 Assume that (H1)-(H4) are satisfied and there exists c 1 ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq167_HTML.gif such that m c 1 M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq168_HTML.gif holds. Then the boundary value problem (1.1), (1.2) has at least three positive solutions if the following conditions hold:

(H5) There exists a constant r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq169_HTML.gif such that
h ( z 1 , z 2 ) < min { 1 , 1 sup t [ 0 , ) c p ( t ) } r B ( r ) 0 G ( s , s ) p ( s ) v ( s ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equad_HTML.gif

for t [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq113_HTML.gif and z 1 [ 0 , r ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq170_HTML.gif, z 2 [ r , r ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq171_HTML.gif;

(H6) There exist L > r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq172_HTML.gif and an interval [ a , b ] [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq173_HTML.gif such that
h ( z 1 , z 2 ) > L γ o k 0 0 G ( s , s ) p ( s ) v ( s ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equae_HTML.gif

for t [ a , b ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq174_HTML.gif and z 1 [ L , K ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq175_HTML.gif, z 2 [ K , K ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq176_HTML.gif;

(H7) There exist 0 < r < L < K < γ 0 k 0 min { 1 , sup t [ 0 , ) c p ( t ) } [ R B ( R ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq177_HTML.gif, N 1 L max { 1 , sup t [ 0 , ) c p ( t ) } K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq178_HTML.gif, where N = min { c 1 , γ 0 k 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq179_HTML.gif such that
h ( z 1 , z 2 ) < min { 1 , 1 sup t [ 0 , ) c p ( t ) } R B ( R ) 0 G ( s , s ) p ( s ) v ( s ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equaf_HTML.gif

for t [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq113_HTML.gif and z 1 [ 0 , R ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq180_HTML.gif, z 2 [ R , R ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq181_HTML.gif.

Proof The conditions of the Leggett-Williams fixed point theorem will be shown to be satisfied. Define the cone P B https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq182_HTML.gif by P = { z B : z ( t ) 0  for each  t [ 0 , ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq183_HTML.gif and the nonnegative, continuous, concave functional φ : P [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq184_HTML.gif by φ ( z ) = min t [ a , b ] | z ( t ) | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq185_HTML.gif.

Then we have φ ( z ) z https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq186_HTML.gif for all z P https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq187_HTML.gif. If z P ¯ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq188_HTML.gif, then z R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq189_HTML.gif and from (H7) we have
| A z ( t ) | = | 0 G ( t , s ) p ( s ) f ( s , z ( s ) , z ( s ) ) d s + φ ( t ) D 0 g 1 ( z ( s ) ) ψ ( s ) d s + θ ( t ) D 0 g 2 ( z ( s ) ) ψ ( s ) d s | 0 G ( s , s ) p ( s ) v ( s ) h ( z ( s ) , z ( s ) ) d s + φ ( 0 ) D 0 g 1 ( z ( s ) ) ψ ( s ) d s + θ ( ) D 0 g 2 ( z ( s ) ) ψ ( s ) d s 0 G ( s , s ) p ( s ) v ( s ) h ( z ( s ) , z ( s ) ) d s + M D [ g 1 ( R ) + g 2 ( R ) ] 0 ψ ( s ) d s R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equag_HTML.gif
Furthermore,
| ( A z ) ( t ) | 0 | G ( t , s ) | p ( s ) v ( s ) h ( z ( s ) , z ( s ) ) d s + a 2 D p ( t ) 0 g 1 ( z ( s ) ) ψ ( s ) d s + a 1 D p ( t ) 0 g 2 ( z ( s ) ) ψ ( s ) d s c p ( t ) 0 G ( s , s ) p ( s ) v ( s ) h ( z ( s ) , z ( s ) ) d s + max { a 1 , a 2 } D p ( t ) [ g 1 ( R ) + g 2 ( R ) ] 0 ψ ( s ) d s sup t [ 0 , ) c p ( t ) [ 0 G ( s , s ) p ( s ) v ( s ) h ( z ( s ) , z ( s ) ) d s + B ( R ) ] R . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equ29_HTML.gif
(4.1)

Therefore, we get A z R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq190_HTML.gif, and this implies that A : P ¯ R P ¯ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq191_HTML.gif.

Now we show that condition (i) of Theorem 4.1 is satisfied. Let z ( t ) = L + K 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq192_HTML.gif for t [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq113_HTML.gif. By the definition of P ( φ , L , K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq193_HTML.gif, z P ( φ , L , K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq194_HTML.gif. Then { z P ( φ , L , K ) : φ ( z ) > L } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq195_HTML.gif. If z P ( φ , L , K ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq194_HTML.gif, then by (H6) we get
φ ( A z ) = min t [ a , b ] ( 0 G ( t , s ) p ( s ) f ( s , z ( s ) , z ( s ) ) d s + φ ( t ) D 0 g 1 ( z ( s ) ) ψ ( s ) d s + θ ( t ) D 0 g 2 ( z ( s ) ) ψ ( s ) d s ) min t [ a , b ] ( 0 G ( t , s ) p ( s ) f ( s , z ( s ) , z ( s ) ) d s ) γ 0 0 G ( s , s ) p ( s ) u ( s ) h ( z ( s ) , z ( s ) ) d s γ 0 k 0 0 G ( s , s ) p ( s ) v ( s ) h ( z ( s ) , z ( s ) ) d s > L . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equah_HTML.gif

Therefore, condition (i) of Theorem 4.1 is satisfied.

If z r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq196_HTML.gif, then by (H5) we have
| A z ( t ) | = | 0 G ( t , s ) p ( s ) f ( s , z ( s ) , z ( s ) ) d s + φ ( t ) D 0 g 1 ( z ( s ) ) ψ ( s ) d s + θ ( t ) D 0 g 2 ( z ( s ) ) ψ ( s ) d s | 0 G ( s , s ) p ( s ) v ( s ) h ( z ( s ) , z ( s ) ) d s + φ ( 0 ) D 0 g 1 ( z ( s ) ) ψ ( s ) d s + θ ( ) D 0 g 2 ( z ( s ) ) ψ ( s ) d s 0 G ( s , s ) p ( s ) v ( s ) h ( z ( s ) , z ( s ) ) d s + M D [ g 1 ( r ) + g 2 ( r ) ] 0 ψ ( s ) d s < r . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equai_HTML.gif
In a similar way as (4.1), we can see that for each t [ 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq197_HTML.gif,
| ( A z ) ( t ) | sup c p ( t ) [ 0 G ( s , s ) p ( s ) v ( s ) h ( z ( s ) , z ( s ) ) d s + B ( r ) ] < r . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equaj_HTML.gif

Hence, condition (ii) of Theorem 4.1 holds.

Finally, we show that condition (iii) of Theorem 4.1 is also satisfied. If z P ( φ , L , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq198_HTML.gif, we get
| A z ( t ) | 0 G ( s , s ) p ( s ) v ( s ) h ( z ( s ) , z ( s ) ) d s + φ ( 0 ) D 0 g 1 ( z ( s ) ) ψ ( s ) d s + θ ( ) D 0 g 2 ( z ( s ) ) ψ ( s ) d s 0 G ( s , s ) p ( s ) v ( s ) h ( z ( s ) , z ( s ) ) d s + M D 0 [ g 1 ( z ( s ) ) + g 2 ( z ( s ) ) ] ψ ( s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equak_HTML.gif
and
| ( A z ) ( t ) | sup c p ( t ) [ 0 G ( s , s ) p ( s ) v ( s ) h ( z ( s ) , z ( s ) ) d s + M D 0 [ g 1 ( z ( s ) ) + g 2 ( z ( s ) ) ] ψ ( s ) d s ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equal_HTML.gif
Hence, we have
A z max { 1 , sup c p ( t ) } [ 0 G ( s , s ) p ( s ) v ( s ) h ( z ( s ) , z ( s ) ) d s + M D 0 [ g 1 ( z ( s ) ) + g 2 ( z ( s ) ) ] ψ ( s ) d s ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equam_HTML.gif
Therefore, for z P ( φ , L , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq199_HTML.gif and A z > K https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq200_HTML.gif, we have
φ ( A z ) = min t [ a , b ] ( 0 G ( t , s ) p ( s ) f ( s , z ( s ) , z ( s ) ) d s + φ ( t ) D 0 g 1 ( z ( s ) ) ψ ( s ) d s + θ ( t ) D 0 g 2 ( z ( s ) ) ψ ( s ) d s ) γ 0 0 G ( s , s ) p ( s ) f ( s , z ( s ) , z ( s ) ) d s + φ ( ) D 0 g 1 ( z ( s ) ) ψ ( s ) d s + θ ( 0 ) D 0 g 2 ( z ( s ) ) ψ ( s ) d s γ 0 0 G ( s , s ) p ( s ) u ( s ) h ( z ( s ) , z ( s ) ) d s + m D 0 [ g 1 ( z ( s ) ) + g 2 ( z ( s ) ) ] ψ ( s ) d s γ 0 k 0 0 G ( s , s ) p ( s ) v ( s ) h ( z ( s ) , z ( s ) ) d s + c 1 M D 0 [ g 1 ( z ( s ) ) + g 2 ( z ( s ) ) ] ψ ( s ) d s N [ 0 G ( s , s ) p ( s ) v ( s ) h ( z ( s ) , z ( s ) ) d s + M D 0 [ g 1 ( z ( s ) ) + g 2 ( z ( s ) ) ] ψ ( s ) d s ] N 1 max { 1 , sup t [ 0 , ) c p ( t ) } A z > N K max { 1 , sup t [ 0 , ) c p ( t ) } L . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equan_HTML.gif
Therefore, condition (iii) is also satisfied. Then the Leggett-Williams fixed point theorem implies that A has at least three positive solutions z 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq201_HTML.gif, z 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq202_HTML.gif, and z 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_IEq203_HTML.gif which are solutions to the problem (1.1)-(1.2). Furthermore, we have
z 1 < r , z 2 { z P ( φ , L , R ) : φ ( z ) > L } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equao_HTML.gif
and
z 3 P ¯ R { P ( φ , L , R ) P ¯ r } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-127/MediaObjects/13661_2012_Article_298_Equap_HTML.gif

 □

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Ege University

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© Yoruk and Hamal; licensee Springer. 2012

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